On the AVD-total chromatic number of 4-regular circulant graphs

An \textit{AVD-$k$-total coloring} of a simple graph $G$ is a mapping $\pi:V(G) \cup E(G) \to \{1,\ldots,k\}$, with $k \geq 1$ such that: for each pair of adjacent or incident elements $x,y \in V(G) \cup E(G)$, $\pi(x) \neq \pi(y)$; and for each pair of adjacent vertices $x,y \in V(G)$, sets $\{\pi(x)\} \cup \{\pi(xv) \mid xv \in E(G)  \text{ and } v \in V(G)\}$ and  $\{\pi(y)\} \cup \{\pi(yv) \mid yv \in E(G) \text{ and } v \in V(G)\}$ are distinct. The \textit{AVD-total chromatic number}, denoted by $\chi''_{a}(G)$ is the smallest $k$ for which  $G$ admits an AVD-$k$-total-coloring. In 2005, Zhang et al. conjectured that any graph $G$ has $\chi''_{a}(G) \leq \Delta+3$, where $\Delta$ is the maximum degree of $G$ and this conjecture is known as AVD-Total Coloring Conjecture (AVD-TCC). In this article, we determine that the AVD-total chromatic number of $C_n(a,b)$ is $6$, where $n$ is even and $a$, $b$ are both odd such that $1 \leq a < b <\lfloor n^{}/2\rfloor$,  and $C_n(1,k)$ is $6$,   where $n$ and $\ell = \frac{n}{\gcd(n,k)}$ are even.